Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations

In this paper we consider 2 x 2 matrix functions analytic on the upper half plane associated to the hypergeometric function of Gauss, and establish transformations of these matrix functions under some modular transformations. The matrix functions studied here are obtained as the lifts of the local solutions of the matrix hypergeometric differential equation of SL type (i.e., whose image of monodromy representation is contained in S£(2, C)) at 0, 1, oo to the upper half plane by the lambda function (§2). Each component of the matrix functions is represented by a definite integral with a power product of theta functions as integrand. Such an integral was invented by Wirtinger in order to uniformize the hypergeometric function of Gauss to the upper half plane ([5)). In this paper we call it Wirtinger integral (cf. (1.2)). In spite of many possibilities of application of the Wirtinger integral, there seems to be very few examples of application of the Wirtinger integral in literature. One of the advantages of exploiting the matrix functions above in the study of the hypergeometric function is that the monodromy property and the connection relations of the hypergeometric function are all translated as transformations of those matrix functions under modular transformations of the independent variable (§3). Moreover we can derive such transformations by exploiting classical formulas of theta functions without need to use any monodromy property or connection formula of the hypergeometric function. That is to say, this gives another new derivation of the monodromy property and the connection formulas of the hypergeometric function of Gauss

Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients

We consider deformations of $2\times2$ and $3\times3$ matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don't satisfy the well-known system of Schlesinger equations (or its natural generalization). Some general statements concerning reducibility of such deformations for $2\times2$ ODEs are proved. An explicit example of the general non-Schlesinger deformation of $2\times2$-matrix ODE of the Fuchsian type with 4 singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromy and non-isomonodromy deformations of $3\times3$ matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.

Twisted homology and cohomology groups associated to the Wirtinger integral

The first half of this paper deals with the structure of the twisted homology group ssociated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing f the other homology groups is proved (Theorem 1). The second half deals with the structure of the wisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted ohomology groups and the cohomology groups associated to a subcomplex of the de Rham complex is stablished, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted ohomology group) is given (Theorem 2)

Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations

In this paper we consider 2 x 2 matrix functions analytic on the upper half plane associated to the hypergeometric function of Gauss, and establish transformations of these matrix functions under some modular transformations. The matrix functions studied here are obtained as the lifts of the local solutions of the matrix hypergeometric differential equation of SL type (i.e., whose image of monodromy representation is contained in S£(2, C)) at 0, 1, oo to the upper half plane by the lambda function (§2). Each component of the matrix functions is represented by a definite integral with a power product of theta functions as integrand. Such an integral was invented by Wirtinger in order to uniformize the hypergeometric function of Gauss to the upper half plane ([5)). In this paper we call it Wirtinger integral (cf. (1.2)). In spite of many possibilities of application of the Wirtinger integral, there seems to be very few examples of application of the Wirtinger integral in literature. One of the advantages of exploiting the matrix functions above in the study of the hypergeometric function is that the monodromy property and the connection relations of the hypergeometric function are all translated as transformations of those matrix functions under modular transformations of the independent variable (§3). Moreover we can derive such transformations by exploiting classical formulas of theta functions without need to use any monodromy property or connection formula of the hypergeometric function. That is to say, this gives another new derivation of the monodromy property and the connection formulas of the hypergeometric function of Gauss

Linear differential relations satisfied by Wirtinger integrals

We will derive linear differential relations satis ed by Wirtinger integrals by exploiting classical formulas of Jacobi's theta functions, forgetting that Wirtinger integrals are related to Gauss hypergeometric functions, although these linear differential relations are related to ones satis ed by Gauss hypergeometric functions

Linear differential relations satisfied by Wirtinger integrals

We will derive linear differential relations satis ed by Wirtinger integrals by exploiting classical formulas of Jacobi's theta functions, forgetting that Wirtinger integrals are related to Gauss hypergeometric functions, although these linear differential relations are related to ones satis ed by Gauss hypergeometric functions